Finite-Resolution Position Measurements and Their Sharp Limit Detector POVMs, Instruments, and the MTT Completion Boundary

Measurement is an ordinary physical interaction. Its mathematical description nevertheless has several layers: an effect-valued measure gives outcome probabilities, an instrument also gives conditional post-interaction states, a realized record requires an outcome-completion rule, and later dynamics may stabilize that record. This paper constructs the first two layers exactly for finite-resolution position detection and keeps the latter two distinct. On the square-integrable Hilbert space over Euclidean position space, a normalized detector response defines a commutative positive operator-valued measure that is precisely a classical smearing of the sharp position projection-valued measure. Its outcome density is the convolution of the detector response with the sharp Born density, and converges in total variation to that density. For densities with one integrable weak derivative, an explicit error bound separates detector resolution, detector shape, and spatial variation of the input. A square-root instrument realizes the same effects and produces a finite post-measurement filter. For a Gaussian response, detector blur adds exactly epsilon squared times the identity to the outcome covariance. A two-outcome qubit example shows that finite unsharp effects converge in operator norm to a discrete projective measurement, so not every projective measurement is a distributional singularity. In Modal Triplet Theory, these results supply a rigorous downstream detector interface. They do not derive the instrument from selected geometry, select one realized outcome, or identify detector width with a coherence or Ornstein-Uhlenbeck floor.